b Topological Vector Spaces - WSEAS · vector spaces but is included in s topological vector...
Transcript of b Topological Vector Spaces - WSEAS · vector spaces but is included in s topological vector...
b Topological Vector Spaces
RAJA MOHAMMAD LATIF Department of Mathematics and Natural Sciences
Prince Mohammad Bin Fahd University P.O. Box 1664 Al Khobar
KINGDOM OF SAUDI ARABIA [email protected]; [email protected] & [email protected]
https://www.pmu.edu.sa/profiles/rlatif/home
Abstract: - The purpose of the present paper is to introduce the new class of b topological vector spaces. We
study several basic and fundamental properties of b topological and investigate their relationships with certain existing spaces. Along with other results, we prove that transformation of an open (resp. closed) set in a
b topological vector space is b open (resp. closed). In addition, some important and useful
characterizations of b topological vector spaces are established. We also introduce the notion of almost
b topological vector spaces and present several general properties of almost b topological vector spaces.
Key-Words: - b open set, b closed set, topological vector space b topological vector space, almost b topological vector space
Received: February 13, 2020. Revised: April 16, 2020. Accepted: April 24, 2020. Published: April 27, 2020.
1 Introduction
Functional analysis in its traditional sense deals primarily with Banach spaces and, in particular, Hilbert spaces. However, many classical vector spaces have natural topologies not given by the norm. Such, for example, are many spaces of smooth, holomorphic and generalized function. The theory of topological vector spaces is the science of spaces of precisely this kind. The concept of topological vector spaces was first introduced and studied by Kolmogroff [19] in 1934. Since then, different researchers have explored many interesting and useful properties of topological vector spaces. Due to nice properties, topological vector spaces earn a great importance and remain a fundamental notion in fixed point theory, operator theory, and various advanced branches of Mathematics. Now a days the researchers not only make use of topological vector spaces in other fields for developing new notions but also strength and extend the concept of topological vector spaces with every possible way, making the field of study a more convenient and understandable. Recently, Khan et al [17] defined s topological vector spaces as a generalization of
topological vector spaces. Khan and Iqbal [18], in 2016, put forth the concept of irresolute topological vector spaces which is independent of topological vector spaces but is included in s topological vector
spaces. Ibrahim [15] introduced the study of topological vector spaces. In 2018, Sharma et al
[31] introduced and studied another class of spaces, namely almost pre topological vector spaces. In [28, 31, 32, 33} the Ram et al introduced the topological
vector spaces and pre topological vector spaces, almost topological vector spaces and almost
s topological vector spaces and studied properties and
characterizations of these spaces. In 2018, Noiri et al [26] introduced and investigated a new class of sets called b open sets which is weaker than open sets and b open sets. In the present paper, we
introduce and investigate several properties and characterizations of b topological vector spaces.
The relationship of b topological vector spaces
with certain types of spaces is investigated as well. We will also introduce and study almost b topological
vector spaces.
2 Preliminaries
Throughout this paper, ,X and ,Y (or simply,
X and Y ) denote topological spaces on which no separation axioms are assumed unless explicitly stated.
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For a subset A of a space , ,X ,Cl A Int A and
X A denote the closure of ,A the interior of A and the complement of A in ,X respectively. Recently, as generalization of closed sets, the notion of closed sets were introduced and studied by Hdeib [13]. Let
,X be a topological space and let A be a subset of
.X A point x of X is called a condensation point of A if for each open set U with ,x U the set U A is
uncountable. A subset A is said to be closed [7] if it contains all its condensation points. The complement of an closed set is said to be an open set. It is
well known that a subset W of a space ,X is
open if and only if for each x W , there exists an
open set U such that x U and U W is countable. The family of all open subsets of a topological
space , ,X is denoted by , ,O X forms a
topology on X which is finer than . The set of all
open sets of ,X containing a point x X is
denoted by , .O X x The complement of an
open set is said to be an closed set. The
intersection of all closed sets of X containing A is called the closure of A and is denoted by .Cl A
The union of all open sets of X contained in A is
called interior of A and is denoted by .Int A A
subset A of a topological space X is said to be
b open if .A Int Cl A Cl Int A U The
complement of a b open set is called a b closed set.
The intersection of all b closed sets containing A is called the b closure of A and is denoted by .bCl A
The union of all b open sets of X contained in A is
called the b interior A and is denoted by .bInt A
Definition 2.1. A subset A of a space X is said to be
b open if for every ,x A there exists a b open
subset xU X containing x such that xU A is
countable. The complement of an b open subset of
X is called an b closed subset of .X The family of all b open sets in a topological space
,X is denoted by ,b O X or .b O X
The family of all b closed sets in a topological
space ,X is denoted by ,b C X or
.b C X
For any ,x X we present ,b O X x
: .U X x U and U is b open in X
Lemma 2.2. For a subset of a topological space, both
openness and b openness imply b openness.
Proof . 1 Assume A is open, then for each
,x A there is an open set xU containing x such that
xU A is a countable set. Since every open set is
b open, A is b open.
2 Let A be b open. For each ,x A there exists
a b open set open set xU A such that xx U and
.xU A Therefore, A is b open. The following diagram shows the implications for properties of subsets
open set b open set
open set b open set
The converses need not be true as shown by the examples by Noiri et al [26] in 2008. Lemma 2.3. A subset A of a space X is b open if
and only if for every ,x A there exists a b opensubset U containing x and a countable subset C such that .U C A
Lemma 2.4. Let ,X be a topological space.
1 The intersection of an open set and a b open set is
b open set.
2 The union of any family of b open sets is a
b open set.
Proposition 2.5. The intersection of an b open set
and a b open set is b open.
Proposition 2.6. The intersection of an b open set
with an open set is b open.
The intersection of two b open sets is not always b open.
Example 2.7. Let X ΅ with the usual topology .
Let A Q be the set of rational numbers and [0, 1).B
Then A and B are b open, but A BI is not
b open, since each b-open set containing 0 is uncountable set. Proposition 2.8. The union of any family of
b open set is b open.
Proof . Let :A be a collection of b open
subsets of a space X. Then for any ,x AU there
exists such that .x A Hence there exists a
b open subset U of X such that U A is
countable. Now because ,U A U A U and
hence U AU is countable. Therefore,
x AU is b open.
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The intersection of all b closed sets of X
containing A is called the b closure of A and is denoted by .bCl A The union of all b open sets
of X contained in A is called the b interior of A and is denoted by .bInt A
Definition 2.9. A function : , ,f X Y is
said to be b continuous if for each open set W in ,Y the inverse image 1 .f W b O X
Equivalently a mapping : , ,f X Y is called
b continuous if for each x X and each open set V
in Y containing f x , there exists an b open set
U in X containing x such that f U V .
Definition 2.10. Let X be a vector space over the
field K , where K or ΅ £ with standard
topology. Let be a topology on X such that the following conditions are satisfied:
1 . For each x, y X and each open set W X
containing x y, there exist open sets U and V in
X containing x and y respectively, such that U V W ,
2 . For each K, x X and each open set
W X containing x, there exist open sets U in
K containing and V in X containing x such that U.V W .
Then the pair X , is called a topological vector
space. 3 b Topolog ical Vector Spaces
Definition 3.1. Let X be a vector space over the
field K , where K or ΅ £ with standard topology.
Let be a topology on X such that the following conditions are satisfied:
1 . For each x, y X and each open set W X
containing x y, there exist b open sets U and V
in X containing x and y respectively, such that
U V W ,
2 . For each K, x X and each open set W X
containing x, there exist b open sets U in K
containing and V in X containing x such that
U.V W . Then the pair X , is called an
b topogical vector space. (written in short,
b TVS ).
Example 3.2. Let X ΅ be the vector space of real
numbers over the field K , ΅ where X K ΅ is
endowed with standard topology. Then KX , is a
b topogical vector space.
Obviously. every topological vector space is an b topogical vector space but the converse is not
always true. The following are examples of b topogical vector spaces which are not topological
vector space.
Example 3.3. Consider the field K ΅ with standard
topology. Let X ΅ be endowed with the topology
,D, , ΅ where D denotes the set of irrational
numbers. We show that X ,΅ is an b TVS. For
this purpose, we have to prove the following:
1 Let x,y X . If x y is rational, then the only
open set in X containing x y is .΅ So, there is
nothing to prove.
If x y is irrational, then for open neighbourhood
W D of x y in X , we have following cases:
Case i . If both x and y are irrational, we can
choose b open sets U x and V y in X such
that U V W.
Case ii .If one of x or y is rational, say y. Then,
for the selection of b open sets U x and
V p,y in X , where p D such that p x D, we
have U V W.
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This proves the first condition of the definition of b topologoical vector space.
2 Let ΅ and .x X If x is rational, then
verification is straightforward.
Suppose x is irrational. Let W D be an open neighbourhood of .x Then the following cases arise:
Case i . If and x are irrational, then, choose
b open sets ,U Q I U in ΅ and
V x in ,X we have U.V W.
Case ii . If is rational and x is irrational, then for
the selection of b open sets ,U Q I
in ΅ and V x in ,X we have U.V W.
Case iii . Finally, suppose is irrational and x is
rational. Then for the choice of b open sets, take
,U D I in ΅ and ,V x p in ,X
where the selection of p D is such that .p q is
irrational for each q U . Then we obtain U.V W.
Hence X ,΅ is a b topological vector space.
Example 3.4. Let X ΅ be a vector space of real
numbers over the field K ΅ and let be a topology
on X induced by open intervals ,a b and [1, )c
where , , .a b c ΅ We observe that ,΅΅ is an
b topological vector space over the field ΅ with the
topology defined on .΅ We note that for all ,x y ΅
and each open neighbourhood W of x y in ,X there
exist b open neighbourhoods U and V of x and y
respectively in X such that .U V W
Also for each , ΅ x ΅ and for each open
neighbourhood W of x in ,X there exist b open
neighbourhoods U of in ΅ and V of x in X such
that . .U V W However, ,΅΅ is not a topological
vector space because, for instance if, we choose 3x , 4y and an open neighbourhood [1, 2)W
of x y in ,X we can not find open neighbourhoods
U and V containing x and y respectively in X which
satisfy the condition U.V W.
Theorem 3.5. Let KX , be an b topological
vector space. Suppose :xT X X is a translation
mapping defined by xT y y x for each y X
,where x is any fixed element of X and M : X X
is a multiplication mapping defined by M x x,
for each x X 0 is a fixed scalar . Then prove
that xT and xM both are b continuous.
Proof . Let y be an arbitrary element in X and let W
be an open neighbourhood of .xT y y x By
definition of b topological vector spaces, there exist
b open neighbourhoods U and V containing y and
x respectively, such that .U V W In particular, we
have U x W which means .xT U W The
inclusion shows that xT is b continuous at y.
Hence xT is b continuous on .X Now we prove the
statement for multiplication mapping. Let K and .x X Let W be an open neighbourhood of
. .M x x By definition of b topological vector
spaces, there exist b open neighbourhoods U and
V containing and x respectively, such that . .U V W In particular, we have . ,V W which
means .M V W The inclusion shows that xM is
b continuous at .x Hence M is b continuous
on .X
Theorem 3.6. Let A be any open subset of an
b topogical vector space X , . Then the
following statements are true:
1 . x A b O X for each x X ,
2 . .A b O X for each non-zero scalar .
Proof . 1 . Let y x A. Then there exists an a A
such that y x a. Hence, a x y A. Since A
is open set in X. Thus there exist b open sets
U ,V b O X containing x and y, respectively
such that U V A. Therefore, x V A
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V x A y b Int x A
x A b Int x A . Hence it follows that
x A b Int x A . This proves that x A is an
b open set in X .
2 . Let x A. Then x a for some a A. Hence
1a x A
and A is an open set in X . So by the
definition of b topological vector spaces, there exist
b open sets U in containing 1
and V in X
containing x such that U .V A. Whence we find that
x V .A. This shows that x bInt A . Since
x .A was arbitrary, it follows that
.A bInt .A . Thus, .A b O X .
Corollary 3.7. Let A be any non-empty open subset
of a b topological vector space X . Then the
following statements are true:
i . x A Int Cl Int x A for each x X .
ii . .A Int Cl Int .A for each non-zero
scalar .
Corollary 3.3. If U is any non-empty open set and B
is any set in an b topological vector space X , then
B U b O X .
Theorem 3.4. Let X , be an b topological
vector space and let F be a closed subset of X . Then the following statements are true:
1 . x F b C X for each x X ,
2 . F b C X for each non-zero scalar .
Proof . 1 . Suppose that y b Cl x F .
Consider z x y and let W be any open set in X
containing z. Then there exists b open sets U and
V in X such that x U , y V and U V W .
Since y b Cl x F , x F V . I So, there is
an a x F V . I Now,
x a F U V F W I I F W I
z Cl F F y x F. It follows that
x F b Cl x F . This proves that x F is
b closed set in X .
2 . Assume that x b Cl .F and let W be any
open set in X containing 1
x.
Since X is
b topogical vector space, there exist b open
sets U in K containing 1
and V in X containing x
such that U .V W . Since x b Cl .F , there
exists a V .F . I This implies that
1 1a F U .V F W
I I F W I
1x Cl F F
x F. It proves that F is
b closed in X .
Theorem 3.5. Let A and B be any subsets of a
b topogical vector space X . Then
b Cl A b Cl B Cl A B .
Proof . Let z b Cl A b Cl B . Then
z x y for some x b Cl A and
y b Cl B . Let W be any open neighbourhood
of z in X . By definition of b topogical vector
spaces, there exist b open sets U and V in X
containing x and y respectively, such that U V W .
Since x b Cl A and y b Cl B , there are
a A U I and b B V I a b A B U V I
A B W I A B W I and hence
z Cl A B . It follows that
b Cl A b Cl B Cl A B .
Theorem 3.6. Let A and B be any subsets of an
b topogical vector space X . Then
A Int B b Int A B .
Proof . Let z A Int B be arbitrary. Then z x y
for some x A and y Int B . This results in
x z Int B and consequently by definition of
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b TVS , there exist b open sets U and V in X
containing x and z respectively, such that
U V Int B . In particular, x V Int B
V x Int B A B. Since V b O X ,
z V b Int A B z b Int A B .
Thus A Int B b Int A B .
4 Characterizations of b TVS.
Theorem 4.1. Let A be any subset of an
b topogical vector space X . Then the following
assertions are true:
1 . b Cl x A x Cl A for each x X .
2 . x b Cl A Cl x A for each x X .
3 . x Int A b Int x A for each x X .
4 . Int x A x b Int A for each x X .
Proof . 1 . Let y b Cl x A and consider
z x y in X . Let W be any open set in X
containing z. Since X is an b topogical vector
space, there exist b open sets U and V in X
containing x and y respectively, such that
U V W . Since y b Cl x A , there is
a x A V . I Then it follows that
x a x x A U V A U V A W I I I
showing that A W I and hence z Cl A . That is,
y x Cl A . Therefore,
b Cl x A x Cl A .
2 . Let z x b Cl A . Then z x y for some
y b Cl A . Let W be an open neighbourhood of
x y in X . Since X is b topogical vector space,
there exist b open sets U and V in X containing
x and y respectively, such that U V W . Since
y b Cl A , A V .I So, there is a A V I and
thus x a x A U V x A W . I I Hence we
have x A W . I Therefore it implies
z Cl x A . This proves the assertion.
3 . Let y x int A . Then y x a for some
a Int A and hence there exist b open sets U
and V in X containing x and y respectively, such
that U V Int A . Now,
x V U V Int A A implies that V x A.
Since V is b open set in X containing y, we must
have y b Int x A . Thus
x Int A b Int x A .
4 . Let z Int x A . Then z x y for some
y A. Since X is b topogical vector space, there
exist b open sets U and V in X containing x and
y respectively, such that
x V U V Int x A x A implies that
x V x b Int A . Since z x y x V , so
z x b Int A . Therefore it follows that
Int x A x b Int A .
Theorem 4.2. Let A be any subset of a b topogical
vector space X . Then the following assertions are true:
a . b Cl .A .Cl A for each non-zero scalar
,
b . . b Cl A Cl .A for each non-zero
scalar ,
c . .Int A b In .A for each non-zero scalar
,
d . Int .A . b Int A for each non-zero
scalar ,
Proof . a . Assume that x b Cl A and let W
be any open set in X containing 1
x.
Since X is a
b topogical vector space, there exist b open
sets U in K containing 1
and V in X containing x
such that U .V W . Since x b Cl A , there
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exists a V A . I This implies that
1 1a A U .V A W
I I A W I
1x Cl A
x .Cl A . It proves that
b Cl .A .Cl A .
b . Suppose that B A and * 1.
Then, by a .
it follows that * *b Cl B .Cl B
1 1b Cl A .Cl A . b Cl A
Cl .A .
c . Let y x for any x Int A . Then
1x y Int A .
Since X is b topogical vector
space. So there exist b open sets U in K
containing 1
and V in X containing y such that
U .V Int A . Then it implies that
1 1x y V U.V Int A A
y V .A.
Since V is b open set. Hence y b Int A .
Therefore it follows that .Int A b In .A .
d . Suppose that B A and * 1.
Then by (c),
it follows that * *Int B b Int .B
1Int .A b Int A
Int .A
. b Int A .
Theorem 4.3. Let A be any subset of an
b topogical vector space X . Then
b Cl x Cl A x Cl A for each x X .
Proof . Assume y b Cl x Cl A . Consider
z x y and let W be an open neighbourhood of
z X . Then there exist b open neighbourhoods U
and V of x and y in X respectively, such that
U V W . Then x V U V W
V x W . Since y b Cl x Cl A , we have
V x Cl A , I which implies that
x W x Cl A I W Cl A .I Since
W is open, W A .I Hence z Cl A , that is
x y Cl A y x Cl A . Consequently,
b Cl x Cl A x Cl A .
Theorem 4.4. Let A be any subset of an
b topogical vector space X . Then prove that
x Int A b Int x b Int A for each
x X .
Proof . Let y x Int A . Then y x a for some
a Int A . As a result of this, we get b open sets
U and V in X containing x and y respectively,
such that U V Int A . Now
x V U V Int A b Int A . Then it
implies V x b Int A . Since V is b open
set in X such that y V , we have that
y b Int x b Int A proving that
x Int A b Int x b Int A .
Definition 4.5. Let B be a subset of a space X . A
collection iU : i I of b open sets of X is called
a b open cover of B if ii IB U .
U A topological
space X is said to be b compact if every
b open cover of X has a finite sub cover. A subset
B of X is said to be b compact relative to X if
every cover of X by b open sets of X has a finite
sub cover.
Theorem 4.6. Let A be any b compact set in a
b topogical vector space X . Then x A is
compact, for each x X .
Proof . Let iU : i I be an open cover of x A.
Then ii IA x U .
U By hypothesis and
Theorem 3 6. , 0
ii IA x U
U for some finite
subset 0I of I . Hence we find that0
ii Ix A U .
U
Therefore, it follows that x A is compact.
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Theorem 4.7. Let X be an b topological vector
space. Then scalar multiple of any b compact subset of X is compact.
Proof . Suppose that A is a b compact set in X .
Let be any scalar. If 0, we are done. Assume
that 0. We have to show that .A is compact. Let
iU : i I be any open cover of .A. Then
i.A U : i I U 1iA . U : i I
U
1iA U : i I .
U Since iU is open in X , so by
Theorem 3.6, 1
iU
is b open set in X .
Consequently, 1
iU : i I
is an b open cover of
A. But A is b compact , there exists a finite subset
0I of I such that 0
1iA U : i I .
U It implies that
0i.A U : i I . U Hence .A is b compact.
Theorem 4.8. Let X be an b topological vector
space and Y be a topological vector space over the same field K. Let f : X Y be a linear map such that
f is continuous at 0. Then f is b continuous
everywhere.
Proof . Let x be any non-zero element of X and V
be any open set in Y containing f x . Since the
translation of an open set in a topological vector space
is open, V f x is open set in Y containing 0. Since
f is continuous at 0, there exists an open set U in X
containing 0 such that f U V f x .
Furthermore, linearity of f implies that f x U V .
By theorem 3 6. , x U is b open and hence f is
b continuous at x. By hypothesis, f is
b continuous at 0. This reflects that f is
b continuous.
Corollary 4.9. Let X be an b topological vector
space over the field K. Let f : X K be a linear
functional which is continuous at 0. Then the set
F x X : f x 0 is b closed.
5 Almost b Topological Vector Spaces
In this section, we define almost b topological
vector spaces and investigate their relationships with certain other types of spaces. Some general properties of almost b topological vector spaces are also
discussed.
Definition 5.1. Let X be a vector space over the field
K , where K R or C with standard topology. Let be
a topology on X such that the following conditions are satisfied:
1 . For each x, y X and each regular open set
W X containing x y, there exist b open sets U
and V in X containing x and y respectively such that
U V W , and
2 . For each x X and every K and each regular
open set W X containing x, there exist
b open sets U in K containing and V in X
containing x such that U.V W.
Then the pair KX , is called an almost
b topological vector space
Theorem 5.2. Let A be any open subset of an
almost b topological vector space X . Then the
following statements are true:
1 . x A b O X for each x X
2 . A b O X for each non-zero scalar .
Proof . 1 . Let y x A. Then y x a for some
a A. Since A is open set in X , there exists a
regular open set W in X such that a W A. This
implies that x y W . Since X is an almost
b topological vector space, there exist b open
sets U and V in X such that x U , y V and
U V W U V A x V A
V x A. Since V is b open. Therefore
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y b Int x A . This shows that
x A b Int x A . Hence x A b O X .
2 . Let x A be an arbitrary element. Since A is
open set in X , there exists a regular open set W
in X such that 1
x W A.
By definition of almost
b topological vector spaces, there exist b open
sets U in the topological field K containing 1
and V
in X containing x such that U.V W. Then
1V U.V W A
implies that x V A. Since V
is b open. Therefore x b Int A and hence
A b Int A . Thus A is b open in X ;
that is, A b O X .
Corollary 5.3. For any open set A in an almost
b topological vector space X , the following
statements are valid:
1 . x A Cl Int X A for each x X ,
2 . A Cl Int A for each non-zero scalar .
Theorem 5.4. Let A be any open set in an almost
b topological vector space X . Then
A B b O X for any subset B X .
Proof . Follows directly from Theorem 5.2.
Theorem 5.5. Let B be any closed closed set in
an almost b topological vector space X . Then the
following statements are true:
1 . x B b C X for each x X ,
2 . B b C X for each non-zero scalar .
Proof . 1 . We need to show that
x B b Cl x B . For, let y b Cl x B
be an arbitrary element. Fix z x y. Let W be
any open set in X containing z. By definition of
open sets, there is a regular open set G in X such
that z G W. Then there exist U ,V b O X
such that x U , y V and U V G. Since
y b Cl x B , then by definition,
x B V . I Let a x B V . I Then it implies
x a B x V B U V B G B W . I I I I
This shows that B W .I Thus shows that
z x y Cl B . Since B is closed set, we
have z x y B. This implies y x B. Thus
x B b Cl x B . Hence x B b C X .
2 . We have to prove that B b Cl B . For,
let x b Cl B be arbitrary and let W be any
open set in X containing 1 x. By definition,
there is a regular open set G in X such that 1 x G W. Then there exist b open sets U
containing 1 in topological field K and Vcontaining x in X such that U.V G. Since
x b Cl B , then there is an a B V . I Now
1 1a B .V B U .V B G B W . I I I I
This implies that B W .I Therefore
1 x Cl B B. This shows that x B. Hence
B b Cl B . Thus B b C X .
Theorem 5.6. Let A be a subset of an almost
b topological vector space X . Then the following
assertions hold:
1 . x b Cl A Cl x A for each x X ,
2 . b Cl A Cl A for each non-zero
scalar .
Proof . 1 . Let z x b Cl A . Then z x y
for some y b Cl A . Let W be an open set in X
containing z, then z W Int Cl W . Since X is
an almost b topological vector space, then there
exist U ,V b O X containing x and y
respectively such that U V Int Cl W . Since
y b Cl A , then there is some a A V . I Hence
x a x A U V x A Int Cl W . I I
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This implies x A Int Cl W . I Thus
z Cl x A . Therefore it follows that
x b Cl A Cl x A .
2 . Let x b Cl A and let W be an open set in
X containing x. Then x W Int Cl W ,
so there exist b open sets U containing in
topological field K and V containing x in X such that
U .V Int Cl W . Since x b Cl A , then there
is some b A V . I Therefore it follows immediately
b A V A U .V I I
A Int Cl W I A Int Cl W . I This
implies that x Cl A . Consequently, it shows
that b Cl A Cl A .
Theorem 5.7. Let A be a subset of an almost
b topological vector space X . Then the following
statements are true:
1 . b Cl x A x Cl A for each x X ,
2 . b Cl A Cl A for each non-zero
scalar .
Proof . 1 . Let y b Cl x A and let W be an
open set in X containing x y. Since X is an almost
b topological vector space and W Int Cl W ,
there exist b open sets U and V in X such that
x U , y V and U V Int Cl W . Since
y b Cl x A , there is some a x A V I and
hence x a A U V A Int Cl W I I
A Int Cl W I x y Cl A
y x Cl A . Thus b Cl x A x Cl A .
2 . Let x b Cl A and W be an open set in X
containing 1 x. So there exist b open sets U in
topological field K containing 1 and V in X
containing x such that U .V Int Cl W . As
x b Cl .A , .A V . I Let z A V . I
Then we have 1 1z A V A U .V I I
A Int Cl W I A Int Cl W . I Therefore
1 x Cl A and hence x Cl A . Thus
b Cl A Cl A .
Theorem 5.8. Let A be an open set in an almost
b topological vector space X . Then the following
assertions hold:
1 . b Cl x A x Cl A for each x X ,
2 . b Cl A Cl A for each non-zero
scalar .
Proof . 1 . Let y b Cl x A and let W be an
open set in X containing x y. Since X is an almost
b topological vector space and W Int Cl W ,
there exist b open sets U and V in X such that
x U , y V and U V Int Cl W . Since
y b Cl x A , there is an a x A V . I Now
x a A x V A U V A Int Cl W I I I
A Int Cl W I A Cl W .I Since A
is open, A W .I Therefore it follows that
x y Cl A ; that is, y x Cl A . Thus it shows
that b Cl x A x Cl A .
2 . Let y b Cl A and W be an open set in X
containing 1 y. Then there exist b open sets U
in topological field K containing 1 and V in X
containing y such that U .V Int Cl W . As
y b Cl A , there is some b A V . I Thus
1 1b A V A U .V A Int Cl W I I I
A Int Cl W I A Cl W .I Since A is
open, so A W .I Therefore it follows that
1 y Cl A y Cl A . Consequently, we
obtain b Cl A Cl A .
Theorem 5.9. Let A and B be subsets of an almost
b topological vector space X . Prove that
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b Cl A b Cl B Cl A B .
Proof . Let x b Cl A and y b Cl B . Let
W be an open neighbourhood of x y in X . Since
W Int Cl W and Int Cl W is regular open,
there exist U ,V b O X such that x U , y V
and U V Int Cl W . Since x b Cl A and
y b Cl B , there are a A U I and b B V . I So
a b A B U V A B Int Cl W I I
A B Int Cl W . I Thus x y Cl A B ;
That is, b Cl A b Cl B Cl A B .
Theorem 5.10. Let A be any subset of an almost
b topological vector space X . Then the following
assertions are true:
1 . Int x A x b Int A for each x X ,
2 . x Int A b Int x A for each x X ,
Proof . 1 . Let y Int x A . We know that
Int x A is open. Then for each
y Int x A , there exist a regular open set W in
X such that y W Int x A . Since
y Int x A x A , y x a for some a A.
Since X is an almost b topological vector space,
then there exist b open sets U and V in X
containing x and a respectively such that U V W. Thus it follows that
x V W Int x A x A. Hence we have
V x x A A. Since V is b open, then
V b Int A and therefore a b Int A
x y b Int A y x b Int A . It
shows that Int x A x b Int A .
2 . Let y x Int A . Then there exists a regular
open set W in X such that
x y W Int A A. By definition of almost
b topological vector spaces, there exist b open sets U and V in X containing x and y
respectively, such that U V W. Thus
x V U V W V x W x A. Sine V is
b open set. Thus, y b Int x A . Hence it
follows that x Int A b Int x A .
Theorem 5.11. Let A be a subset of an almost
b topological vector space X . Then the following
assertions are true:
1 . Int A b Int A , for any nonzero
scalar .
2 . Int A b Int A , for each nonzero
scalar .
Proof . 1 . Let y Int A . Then there exists a
regular open set W in X such that
y W Int .A . Then y a for some a A. By
definition of almost b topological vector spaces,
there exist b open sets U in the topological field
K containing and V in X containing a respectively, such that U.V W. Now
y a V U .V W Int A A
a V A. Since V is b open set in X . Hence
a b Int A y a b Int A .
Therefore it shows that Int A b Int A ,
2 . Let y Int A . Then y a for some
a Int A and Int A is open in X . Thus
there exists a regular open set W in X such that
1a y W Int A . By definition of almost
b topological vector spaces, there exist b open
sets U in topological field K containing 1 and V in X containing y respectively such that U.V W.
Now 1 1a y V U.V W Int A A
y V A. Since V is b open set. It follows that
y b Int A . Therefore it shows that
Int A b Int A ,
Theorem 5.12. Let X be an almost b topological
vector space. Then the following assertions are true:
1 . The translation mapping xT : X X defined by
xT y x y, x, y X , is almost b continuous.
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2 . The multiplication mapping M : X X defined
by M y x, x X , is almost continuous ,
where is non-zero fixed scalar in K .
Proof . 1 . Let y X be an arbitrary element. Let W
be any regular open set in X containing xT y . Then,
by definition of almost b topological vector spaces,
there exist b open sets U in X containing x and V
in X containing y such that U V W. This results
in x V W xT V W . This indicates that xT
is almost b continuous at y and hence xT is almost
b continuous.
2 . Let x X and W be any regular open set in X
containing x. Then there exist b open sets U in the
topological field K containing and V in X containing x such that U .V W . This gives that
V W. This means that M V W showing that
M is almost b continuous. at x. Since x X was
an arbitrary element, it follows that M is almost
b continuous.
6 Conclusion
Topological vector spaces are a fundamental notion and play an important role in various advanced branches of mathematics like fixed point theory, operator theory, etc. This paper expounds b topological vector
spaces and the almost b topological vector spaces
which are basically a generalization of topological vector spaces. This paper makes us familiar with
b topological vector spaces as well as with the
origin of almost b topological vector spaces and the
elementary concepts that are used to develop the theory of almost b topological vector spaces. We
investigate several new properties and characterizations of b topological vector spaces and almost
b topological vector spaces. We explore some more
basic features of b topological vector spaces and
almost b topological vector spaces and interprets
their relationships with some well-known existing spaces.
Acknowledgment
The author is highly and gratefully indebted to the Price
Mohammad Bin Fahd University, Al Khobar, Saudi
Arabia, for providing necessary research facilities
during the preparation of this paper.
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